%n^" 


Out.-^ 


EXAMPLES 


DIFFERENTIAL    EQUATIONS 


RULES  FOR  THEIR  SOLUTION. 


BY 


GEORGE  A.   OSBORNE,  S.B. 

Professor  op  Mathematics  in  the  Massachusetts  Institute 
OF  Technology. 


o>&<o 


BOSTON,   U.S.A.: 

PUBLISHED  BY  GINN  &  COMPANY. 

189  9. 


Entered,  according  to  Act  of  Congress,  in  the  year  1886,  by 

GEORGE  A.  OSBORNE, 
in  the  Office  of  the  Librariaci  of  Congress,  at  Washington. 


J.  S.  CusHiNG  &  Co.,  Printers,  Bostow. 


06 


PREFACE. 


THIS  work  has  been  prepared  to  meet  a  want  felt 
by  the  author  in  a  practical  course  on  the  subject, 
arranged  for  advanced  students  in  Physics.  It  is  in- 
tended to  be  used  in  connection  with  lectures  on  the 
theory  of  Differential  Equations  and  the  derivation  of 
the  methods  of  solution. 

Many  of  the  examples  have  been  collected  from  standard 
treatises,  but  a  considerable  number  have  been  prepared 
by  the  author  to  illustrate  special  difficulties,  or  to  pro- 
vide exercises  corresponding  more  nearly  with  the  abilities 
of  average  students.  With  few  exceptions  they  have  all 
been  tested  by  use  in  the  class-room. 

G.  A.   Osborne. 

Boston,  Feb.  1,  1886. 


n^30f>150 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

Microsoft  Corporation 


http://www.archive.org/details/examplesofdifferOOosborich 


CONTEI^TS. 


CHAPTER   I. 

DEFINITIONS. DERIVATION   OF    THE    DIFFERENTIAL   EQUATION   FROM    THE 

COMPLETE    PRIMITIVE. 

PAGE. 

Definitions 1 

Derivation  of  differential  equations  of  the  first  order   1 

Derivation  of  differential  equations  of  higher  orders 2 


SOLUTION  OF  DIFFERENTIAL  EQUATIONS. 
CHAPTER  II. 

DIFFERENTIAL   EQUATIONS    OF    FIRST   ORDER   AND    DEGREE    BETWEEN    TWO 

VARIABLES. 

Form,    XYdx  -^-X'Y^dy^O '. 4 

Homogeneous  equations 5 

Form,     (ax  -\-  hy  ■\-  c)  dx  ■\-  (a^x  ■\-h'y  ■\-  d^dy  =  0   5 

Linear  form,    -^  ■\-  Py  =  Q 6 

dx 

Form,    ^+Py=Qy- 7 

dx 

CHAPTER   III. 

EXACT   DIFFERENTIAL   EQUATIONS.  —  INTEGRATING   FACTORS. 

Solution  of  exact  differential  equations   8 


VI  CONTENTS. 

PAGE. 

Solution  by  means  of  an  integrating  factor  in  the  following  cases : 

Homogeneous  equations 9 

Form,    f^{xy)ydx-\-f^{xy)xdy  =  (> 10 

dM_dN  dN      dM 

When    iL_^  =  K-),    or±^=Hy) 10 

CHAPTER  IV. 

DIFFERENTIAL   EQUATIONS   OF   FIRST   ORDER  AND   DEGREE   WITH   THREE 
VARIABLES. 

Condition  for  a  single  primitive,  and  method  of  solution 12 

CHAPTER  V. 

DIFFERENTIAL   EQUATIONS   OF    THE    FIRST    ORDER   OF    HIGHER   DEGREES. 

Equations  which  can  be  solved  with  respect  to  p 14 

Equations  which  can  be  solved  with  respect  to  y 14 

Equations  which  can  be  solved  with  respect  to  a: 15 

Homogeneous  equations 16 

Clairaut's  form,    y  =px  -{-/(^p) 16 

CHAPTER   VI. 

SINGULAR    SOLUTIONS. 

Method  of  deriving  the  singular  solution  either  from  the  complete 

primitive  or  from  the  differential  equation 17 


DIFFERENTIAL  EQUATIONS  OF  HIGHER   ORDERS. 
CHAPTER  VII. 

LINEAR   DIFFERENTIAL   EQUATIONS. 

Linear  equations  with  constant  coefficients  and  second  member  zero. .  19 
Linear  equations  with  constant  coefficients  and  second  member  not  zero,  22 
A  special  form  of  linear  equations  with  variable  coefficients 24 


CONTENTS.  VU 


CHAPTER   VIII. 

SPECIAL   FORMS   OF    DIFFERENTIAL   EQUATIONS   OF    HIGHER   ORDERS, 

PAGE. 

Form,    ^=X 25 

Form,    4^  =  r 25 

Equations  not  containing  y  directly , 26 

Equations  not  containing  x  directly 27 


CHAPTER   IX. 

SIMULTANEOUS   DIFFERENTIAL   EQUATIONS. 

Simultaneous  equations  of  first  order 28 

Simultaneous  equations  of  higher  orders 30 

CHAPTER   X. 
Greometrical  applications 32 

Answers  to  Examples 35 


EXAMPLES  OF    DIFFERENTIAL    EQUATIONS. 


o^t^o 


CHAPTER   L 

DEFINITIONS.      DERIVATION  OF   THE   DIFFERENTIAL  EQUA- 
TION  FROM   THE   COMPLETE   PRIMITIVE. 

1.  A  differential  equation  is  an  equation  containing  differen- 
tials or  differential  coefficients. 

The  solution  of  a  differential  equation  is  the  determination  of 
another  equation  free  from  differentials  or  differential  coeffi- 
cients, from  which  the  former  may  be  derived  by  differentiation. 

The  order  of  a  differential  equation  is  that  of  the  highest 
differential  coefficient  it  contains ;  and  its  degree  is  that  of  the 
highest  power  to  which  this  highest  differential  coefficient  is 
raised,  after  the  equation  is  freed  from  fractions  and  radicals. 

The  solution  of  a  differential  equation  requires  one  or  more 
integrations,  each  of  which  introduces  an  arbitrary  constant. 
The  most  general  solution  of  a  differential  equation  of  the  nth 
order  contains  n  arbitrary  constants,  whatever  may  be  its  de- 
gree. This  general  solution  is  called  the  complete  primitive  of 
the  given  differential  equation. 

2.  To  derive  a  differential  equation  of  the  first  order  from  its 
complete  primitive. 

Differentiate  the  primitive  ;  and  if  the  arbitrary  constant  has 
disappeared,  the  result  is  the  required  differential  equation.  If 
not,  the  elimination  of  this  constant  between  the  two  equations 
will  give  the  differential  equation. 


2  DERIVATION   OF  THE  DIFFERENTIAL  EQUATION. 

3.  Form  the  differential  equations  of  the  first  order  of  which 
the  following  are  the  complete  primitives,  c  being  the  arbitrary 
constant : 

1.  log(xy)  -{-x  =  y  +  c. 

2.  (l+a^)(H-2/')=caj2. 

3.  cosy=ccosx, 

4.  y=z  ce-**"'^*  +  tan-^aj  —  1. 

5.  y={cx-\-logx-\'l)~\ 

6.  y  =  cx-\-c^c^, 

7.  {y-\-cy  =  4:ax. 

8.  y^s'm^x  +  2cy-{-c^  =  0.      . 

9.  e2^  +  2ca;e^H-c2  =  0. 

4.  To  derive  a  differential  equation  of  the  second  order  from 
its  complete  primitive. 

Differentiate  the  primitive  twice  successively,  and  eliminate, 
if  necessary,  the  two  arbitrary  constants  between  the  three 
equations. 

5.  Form  the  differential  equations  of  the  second  order  of 
which  the  following  are  the  complete  primitives,  Ci  and  Cg  being 
the  arbitrary  constants : 


1. 

2/ =  Ci  cos  (ax+Cj). 

2. 

y  =  cie"  +  c^e-". 

3. 

y=(ci  +  C2x)e'". 

4. 

y  =  C^  +  ^- 

6. 

,   cosaa; 

?/  =  Ci  sm  nx  -f  Co  cos  nx  +  — - 

n^  —  or 

DERIVATION   OF  THE  DIFFERENTIAL  EQUATION.         3 

6.  The  preceding  process  may  be  extended  to  the  derivation 
of  equations  of  higher  orders  from  their  primitives. 

7.  Form  the  differential  equations  of  the  third  order  of  which 
the  following  are  the  complete  primitives : 

1.  y  =  Cie^'-]-C2e~^'-\-Cze'. 

2.  2/^*  =  Cie^'^  +  C2sina;V2  +  C3COsajV2. 

3.  y  =  U  4-  C2X  +  —je'  +  C3. 

Form  the  differential  equations  of  the  fourth  order  of  which 
the  following  are  the  complete  primitives : 

4.  y=  (ci  +  Cgi^-F  03x2)6' +  C4. 

6.       oi:^  +  a^y  =  Ci  e"*  +  Cg e"***  +  c^  sin  ax  +  c^  cos  ax. 


CHAPTER    II. 

DIFFERENTIAL    EQUATIONS     OF     THE     FIRST    ORDER    AND 
FIRST   DEGREE   BETWEEN   TWO   VARIABLES. 

General  Form,        Mdx  +  Ndy  =  0, 
where  Jf,  JSf^  are  each  functions  of  x  and  y, 

8.  Form,  XYdx  +  X'Y'dy  =  0, 

where  X,  X',  are  functions  of  x  alone,  and  F,  F',  functions  of 
y  alone. 

Divide  so  as  to  separate  the  variables,  and  integrate  each 
part  separately. 

9.  Solve  the  following  equations  : 

1.  {1 -\-x)ydx-{- (l—y)xdy  =  0, 

2.  (x'  -  yx")  ^  +  2/'  +  xy'  =  0. 

dx 


3. 


dy^_±+jf_, 
dx      {l-\-x'^)xy 


4.        a[xJ--\-2y\  =  xyJ-' 
dx  J  ctx 


5.  {l+f)dx=^(y+^l+f){l+xy2dy. 

6.  sinaJCOS2/cZa;=cosa;sin2/(^2/- 

7.  sec^ajtan2/c?aj  + sec^?/taniC(^2/==  ^• 

8.  sec^a;tan2/d2/H-sec^2/^^^^^^==^' 

9^       dy  I  1+y  +  y'^Q^ 
da;      1  +  a:  +  a;2 


HOMOGENEOUS   EQUATIONS.  6 

10.  Homogeneous  equations. 

Substitute  y  =  vx\  in  the  resulting  equation  between  v  and  a?, 
the  variables  can  be  separated.     (See  Art.  8.)     . 

11.  Solve  the  following  equations  : 

1.  (y  —  x)dy-j-ydx  =  0. 

2.  (2^xy  —  x)dy  -{-ydx  =  0. 

3.  2/^  +  ^— =  ^2/—*  *' 

dx  dx 

4.  x^^y+'y/x'  +  y^ 

dx 

5.  xGos^'—=zyeos^  —  x. 

X  dx  X 

6.  (8y  +  10x)dx+  {oy-i-Jx)dy  =  0. 

7.  (x  +  y)-^  =  y-x, 

8.  xcos^  (ydx-i-xdy)  =  y  sin^  (x  dy  ^  ydx). 

X  X 

9.  x-\-y-^  =  my. 

dx 

(1),  m<2;     (2),m  =  2;     (3),  m>2. 

10.        [(aJ^  —  y^)  sma  +  2xy  cosa  —  y^x'  +  2/T  ^ 
=  2 0^2/  sin  a  —  (a?^  —  2/^)  cosa  +  x  -y/x^  +  2/^. 

12.  Form, 

(aa;  +  6y +  c)di»+  (a'a7  4- ?>'2/ +  ^0^^  =  ^- 

Substitute  a;  =  a;'  +  a,     2/  =  2/'  +  i^? 

and  determine  the  constants  a,  ^,  so  that  the  new  equation  be- 
tween x'  and  y'  may  be  homogeneous.     (See  Art.  10.) 


6  LINEAK  EQUATIONS   OF  FIKST  ORDER. 

This  method  fails  when  —  =  — .    In  this  case  put  ax-\-hy  =  z^ 

and  obtain  a  new  equation  between  x  and  z  or  between  y  and  z ; 
the  variables  can  then  be  separated. 


13.    Solve  the  following  equations  : 

1.  {3y  —  7x-\-7)dx-{-  {7y —  3x-{-S)dy  =  0. 

2.  Ux-\-2y-l)^  +  2x-\-y-{-l=0. 

dx 


3. 


dy  _  7y-\-  x-{-2  ^ 
dx'     3  a;  +  5  2/  +  6 

4.        {2y  +  x-\- 1)  dx  =  (2a;  +  4?/  +  3)dy. 


5.       2x-y  +  l-^(x-hy -2)^=^0. 


14.  Linear  Form,         -^  +  Py  =  Q^ 

dx 

where  P,  Q,  are  independent  of  y. 

Solution,  y  =  e'-^^'^y  i  Qe'^^dx -f  cj- 

15.  Solve  the  following  equations  : 

1.  x-^  —  ay  =  x-\-\. 

dx 

2.  X  {\  ^  x^)dy  -{-  {2x^  —  l)ydx  =  a:x?dx. 


2^. 


3.  (1  -  x'Y^  +  2/  Vl  -  ic'  =  a;  +  Vl  -  a^. 

4.  -^ +'ycosa;  = -sin2a;. 
dx     ^  2 

6.        (1  +  y'^)dx  =  (tan-^2/  —  ^)  ^V- 


EXTENSION   OF   LINEAR   FORM.  7 

\        dxj 

7.  (l+x^)dy-{-fxy--\dx  =  0. 

8.  ^^y^  =  ^^, 

dx        dx         dx 

where  </>  is  a  function  of  x  alone. 

16.  Form,  ^  +  Py=^Qy% 

dx 

where  P,  Q,  are  independent  of  y. 

Divide  by  y'^^  and  substitute  z  =  y~'^^.     The  new  equation 
between  z  and  x  will  be  linear.     (See  Art.  14.) 

17.  Solve  the  following  equations  : 

1.  (1 —0?)-^ —  xy=zax7f. 

dx 

2.  32/2^-a2/3  =  a;-f  1. 

dx 

3.  ^=2xy(ax'f-^l). 

4.  ^(x^f-i.xy)=l. 

6.        — +  2/ cos  a?  =  2/"  sin  2  a?. 
dx 

6.  (y\ogx-~l)ydx  =  xdy. 

7.  aa^y^'dy  -^-ydx  =  2xdy, 

8.  y  — cosaj-^  =2/^cosir  (1  —  sina;). 

dx 

9«       y-^-\-^y^=^cL008x, 
dx 


CHAPTER    III. 

EXACT   DIFFERENTIAL   EQUATIONS   AND   INTEGRATING 
FACTORS. 

18.    Mdx  +  Ndy  is  an  exact  differential  when 


<1) 


dM^  dJSr 
dy       dx 

The  solution  of 

Mdx-\'  Ndy  =  0,     in  this  case  is 

Cmdx  +  ((n-  —  CMdx\dy  =  c, 

or  CNdy  +  (i^-  — ,  f^dy\x  =  c. 

In  integrating  with  respect  to  a?,  y  is  regarded  as  constant, 
and  in  integrating  with  respect  to  2/,  x  is  regarded  as  constant. 

19.    Solve  the  following  equations  after  applying  the  condi- 
tion (1)  for  an  exact  differential : 

1.  (aj3  -f-  3  xy^)  dx  +  (^/^  -f-  3  x^y)  dy  =  0. 

2..  (x^  --ixy  —  2y^)dx  -f  (y^  —  4xy  —  2x^)dy  =  0. 

3.  fl^^\dx-^^-ldy=:0. 

.  2xdx  ,  /I       3a^\,,      ^ 


EXACT    DIFFERENTIAL   EQUATIONS.  9 

6.  _i^_  +  ^l ^       '^^  =  0. 

V?T?      \       -Va^-j-yV  y 

7.  (x  +        ^       Va;  +  (^2/ ^dy  =  0. 

-  -  /         \ 

8.  {l'^e~^)dx-\-e~^ll--]dy  =  0, 

9.  e(x^-\-y^-+'2x)dx-\-2ye''dy=0. 

10.  (m  da?  +  ^  ^2/)  sin  (wa;  +  n?/)  =  (n  dx-\-mdy)  cos  (riaj-f-m^/). 

11.  xdx  +  ydy        ydx  —  xdy^^ 
Vl  +  ar^  -f  2/'         a^  +  2/' 

-o         of dy  —  ayx^'^ dx  , 

(1),  ^>0;     (2),  ^<0  and  =-A;;     (3),  ^  =  0; 
(4),  a  =  0;     (5),  5  =  0. 

20.  When  Mdx-^Ndy  is  not  an  exact  differential,  it  may 
sometimes  be  made  exact  by  multiplying  by  a  factor,  called  an 
integrating  factor.  The  following  are  some  of  the  cases  where 
this  is  possible. 

21.  When  Mdx  +  Ndy  is  homogeneous,  is  an  in- 

Mx  +  Ny 

tegrating  factor.      This  fails  when  Mx  +  Ny  =  0,  but  in  that 
case  the  solution  is  ?/  =  ex, 

22.  Solve  the  following  equations  by  means  of  an  integrating 
factor : 

1.        (a;^ -\-2xy  —  y^)  dx  =^(x^ ^2xy^ y^)  dy. 
X       y         \y       ^J       ' 


10  INTEGRATING   FACTORS. 

4.        a^dx-\-  {Sx^y-\-2f)dy  =  0. 


5.        {x-\/x^  +  y^  —  ^)  dy  -\-  (xy  —  y  -y/x^  +  y^)  dx  =  0 
(See  Art.  11  for  other  examples.) 

23.    Form,         fi{xy)ydx-\-f2{xy)xdy  =  0. 
1 


is  an  integrating  factor.     This  fails  when 


Mx  —  Ny 

Mx  —  Ny  =  0^ 

but  in  that  case  the  solution  is  xy  =  c. 

Another  method  of  solving  is  to  put  xy  =  v^  and  obtain  an 
equation  between  x  and  v  or  between  y  and  v.  The  variables 
can  then  be  separated. 

24.  Solve  the  following  equations  by  means  of  an  integrating 
factor : 

"ps-^l.  (1 +xy)ydx-{'(l —Qcy)xdy  =  0, 

2.  (x^y^-^xy)ydx-{- (x^y^ —  l)xdy  =  0. 

3.  (x^y^  -f  1)  {xdy-{-  ydx)  +  (x^y^  +  xy)  (ydx  —  xdy)  =  0. 

4.  (^/xy  —  l)xdy  —  (-y/xy  -\-l)ydx=0. 
^'  {y -\-y^/xy)dx-\- {x-\-x^/xy)dy  —  0. 

6.  e^^ix^y^ -f- xy) (xdy  +  ydx)  -{-ydx  —  xdy  =  0, 

7.  xy  [1  +  cot  (xy)'](xdy  -{-ydx)  +  ajd^/  —  ydx  =  0. 

dM_dN 

25.  When  ^L^  =  ^(^), 


then  e*'    *  *  is  an  integrating  factor. 


INTEGRATING  FACTORS.  11 

dN     dM 
Or,  when  dx  ^dy  ^^^y^^ 

then  e-^    ^  ^   is  an  integrating  factor. 

26.    Solve  the  following  equations  by  means  of  an  integrat- 
ing factor : 

1.        (aj2  ^ 2/H  '^x)dx  +  2ydy  =  0. 


3. 


(SQ^-y')^  =  2xy. 

dy^x^-h  y\ 
dx        2xy 


4.  [(1  - 2/)  Vl  -:^-xy'\dx+  [1  - x^ -- x VH^''] dy  =  0 , 

5.  (cosaj+22/sec2/sec2  2a;)da;-f  (tan2a;sec2/— sina7tan2/)(^^=0. 

6.  sin(3 X  -  2y)(2 dx  —  dy)  +  sin  {x  -  2y)dy  =  0. 


7.       The  Linear  Equation 

dx 
where  P  and  Q  are  independent  of  y. 


i+^»=«- 


CHAPTER    IV. 

DIFFERENTIAL    EQUATIONS    OF    THE    FIEST    ORDER    AND 
DEGREE   CONTAINING    THREE   VARIABLES. 

General  form,         Pdx  -f  Qdy  -f-  Rdz  =  0, 
where  P,  Q,  i^,  are  each  functions  of  a?,  2/,  z. 

27.  If  the  variables  can  be  separated,  solve  by  integrating 
the  parts  separately. 

The  equation  is  derivable  from  a  single  primitive  only  when 
the  following  condition  is  satisfied : 

\dz       dy  J         \  dx       dz  J  \dy        dx  J 

The  solution  may  then  be  effected  b^'  first  solving  the  equa- 
tion with  one  of  the  parts  Pdx^  Qdy,  Rdz,  omitted,  regarding 
X,  y,  z,  respectively,  constant. 

Omitting  Rdz,  for  example,  we  solve  Pdx -\-  Qdy  =  0,  re- 
garding z  constant,  and  introducing  instead  of  an  arbitrary 
constant  of  integration,  Z,  an  undetermined  function  of  2;,  which 
must  be  subsequently  determined  so  that  this  primitive  may 
satisfy  the  given  differential  equation.  The  equation  of  condi- 
tion for  determining  Z  will  ultimately  involve  only  Z  and  z. 

28.  Solve  the  following  equations  after  applying  the  condi- 
tion (1)  for  a  single  primitive  : 

1.       ^^    I    ^^y   I   ^^  =0. 

x  —  a     y  —  b     z  —  c 
2.        (ic  —  3  2/  —  z)dx  -\-{2y  —  '^x)dy-{-{z  —  x)dz  =  0. 


EQUATIONS   CONTAINING   THRE   VARIABLES.  13 

3.  (y  +  z)dx-]-(z-h  x)dy  -\-{x-\-  y)d^  0. 

4.  yzdx-{-zxdy  -{-xydz  =  0. 

5.  (y  +  z)dx-\-dy -{'dz  =  0. 

6 .  ay^z"^  dx  -\- bz-  x^  dy  -{-  cay^ y^ dz  =  0. 

7 .  zydx  =  zx  dy  -f  y^  dz. 

8.  (ydx-{-xdy)(a-\-z)  =  xydz, 

9.  (y -{-aydx-\-zdy  =  (y -\-a)dz, 

10.  (y^  +  2/^)  do;  4-  (xz  +  2;^)  d?/  +  (/  —  a;j  d^:  =  0. 

11.  (2x^  +  2xy  -\-  2 xz^ -\-  l)dx  -\-  dy  +  2dz  =  0. 

12.  {x^y  —  y^—y^z)dx-\'  (xy^—x^z—x^)u+  {xy^-{-x^y)dz  =  0. 


CHAPTER    V. 

DIFFERENTIAL    EQUATIONS    OF    THE    FIKST   ORDER,   OF   A 
DEGREE  ABOVE   THE   FIRST. 

In  what  follows,  p  denotes  -^• 

dx 

29.  When  the  equation  can  he  solved  with  respect  to  p. 

The  different  values  of  p  constitute  so  many  differential  equa- 
tions of  the  first  degree,  which  must  be  solved  separately,  using 
the  same  character  for  the  arbitrary  constant  in  all. 

If  the  terms  of  each  of  these  separate  primitives  be  transposed 
to  the  first  member,  the  product  of  these  first  members  placed 
equal  to  zero  will  be  the  complete  primitive. 

30.  Solve  the  following  equations  : 

1.  i)2_5^^e^0. 

2.  a?p^-a'  =  Q. 

3.  a;p2_a  =  0. 

4.  xp^  =  1  —  cc. 

6.  x^p^  +  ^xyp  +  ^y'^^O. 

7.  p^  -\-2  xp'^  —  y'^p^  —  2  xy^p  =  0. 

8.  p^  —(^  -^xy-\-  y^)p^  +  {x^y  -\-  ^y'^  -f  xf^p  —  x^y^  =  0. 

9.  p^  +  2py  cot  a;  =  y^* 

31.  When  the  equation  can  he  solved  loith  respect  to  y. 
Differentiate,  regarding  p  variable  as  well  as  x  and  y^  and 


EQUATIONS   SOLVABLE   WITH   RESPECT   TO  X  OR  y.      15 

substitute  for  dy^  pdx.  There  will  result  a  differential  equation 
of  the  first  degree  between  x  and  p.  Solve  this  equation,  and 
eliminate  j>  between  its  primitive  and  the  given  equation. 


32.   Solve  the 

following  equations : 

1. 

x  —  yp  =  ap'. 

2. 

y  =  xf-\-  2p. 

3. 

{x  +  ypy  =  a\\+p') 

4. 

y  =  xp+p-p\ 

5. 

(y-apy=^l+p\ 

6. 

y  =  ap  +  hp^. 

7. 

»?  +  y=p'^. 

8. 

f=o?(\+p^). 

9. 

y=p^  +  2pK 

33.    When  the  equation  can  be  solved  with  respect  to  x. 
Differentiate,  regarding  p  variable  as  well  as  x  and  ?/,  and 

substitute  for  dx,  _Z.     There  will  result  a  differential  equation 
p 

of  the  first  degree  between  y  and  j9.     Solve  this  equation,  and 

eliminate  p  between  its  primitive  and  the  given  equation. 


:.    Solve  the  following  equations  : 

1. 

p^y  +  2px  =  2/. 

2. 

x=p  +  logp. 

3. 

p\x^-{-2ax)  =  a\ 

4. 

x'p^=l-{-p\ 

5. 

(x  —  apy=l+p^; 

also  when  a=  1. 

6. 

x  =  ap-{-  bp^. 

7. 

my  -  nxp  =  yp\ 

16      HOMOGENEOUS   EQUATIONS.  —  CLAIRAUT's   FORM. 

35.  When  the  equation  is  homogeneous  with  respect  to  x 
and  y. 

Substitute  y  =  vx.  If  the  resulting  equation  between  p  and 
V  can  be  solved  with  respect  to  v^  the  given  equation  comes 
under  Art.  31  or  Art.  33. 

But  if  we  can  solve  with  respect  to  p,  substitute  forp,  v  +  a;—, 

dx 

and  there  will  result  a  differentig-l  equation  of  the  first  degree 
between  v  and  x, 

36.  Solve  the  following  equations  : 

1.  xy\p^-{-2)  =2py^-\-i^. 

2.  {2p-^l)xiy=z  a;  V  +  ^V^- 

3.  4:X^  =  3{Sy—px){y-{-px). 

4.  ds  =  (^ydx-\-  (—)   dy,  where  ds  =  Vl  -i-p^  •  dx. 

5.  (nx+pyy  =  {l-^p^){y^-{-nx^). 

37.  Clair aufs  Form^ 

y=px+f(p). 

The  solution  is  immediately  obtained  by  substituting  p=.c. 

38.  Solve  the  following  equations  : 

1.  y=px  +  -' 

2.  y  =px  -\-  2^  —  p^* 

3.  y'  -2pxy  -1  =zp\l  -x"). 

4.  y  =  2p)x-{-  y^py^'  ^^^^  if  =  2/'« 

5.  ayp^  -\-{2x  —  b)p  =  y.  Put  y^  =  y\ 

6.  x^{y  —px)  =  yp^.  Put  y^  =  y\  x^  —  x\ 

7.  e'^=^(2)  — 1)+F^e^''  =  0.  Pute'^  =  a;',  6^  =  2/'. 

8.  (paj  —  y)  (j9y  +  a;)  =  ^^jp.       Put  y^  =  y\  a^  =  »'. 


CHAPTER   VI. 

SINGULAR   SOLUTIONS. 

39.  A  singular  solution  of  a  differential  equation  is  a  solution 
which  is  not  included  in  the  complete  primitive.  Differential 
equations  of  the  first  degree  have  no  singular  solution.  Those 
of  higher  degrees  may  have  singular  solutions,  which  may  be 
derived  either  from  the  complete  primitive,  or  directly  from 
the  differential  equation. 

40.  Let  f{x^  y^  c)=  0   be  the  complete  primitive. 

By  differentiating,   regarding  c  as  the  only  variable,  obtain 

-^  =  0.     If  we  eliminate  c  between  this  equation  and  the  prim- 
dc 

itive,  the  result  will  be  a  singular  solution,  provided  it  satisfies 

the  given  differential  equation. 

41.  Let  /(cc,  2/,  p)  =  0   be  the  given  differential  equation. 
By  differentiating,  regarding  p  as  the  only  variable,  obtain 

S.  =  0.    If  we  eliminate  p  between  this  equation  and  the  given 
dp 

differential   equation,    the  result   will   be    a   singular   solution, 

provided  it  satisfies  the  differential  equation. 

42.  Derive  the  singular  solution  of  the  following  equations, 
directly  from  the  given  equation,  and  also  from  the  complete 
primitive : 

1.  y=rpx-\-—' 

2.  y^'-2xyp-\-(l+x^)p^=l. 


SINGULAR   SOLUTIONS. 

3.  y  — 4a:2/i)+8i/2  =  0.     Futy  =  z\ 

4.  y  =  (x-l)p-^p\ 

5.  y(l+p^)  =  2xp. 

6.  x^p^  '-2(xy^2)p-^y^  =  0. 

7.  (y  —  xp)  (mp  —  n)=  mnp. 


DIFFERENTIAL    EQUATIONS  OF    AN  ORDER 
HIGHER    THAN   THE  FIRST. 

CHAPTER    VII. 

LIKEAR  DIFFERENTIAL  EQUATIONS. 
General  Form, 

the  coefficients  Xi,  Xg,  ...X„  and  X  being  functions  of  x  alone 
or  constants. 

43.    Linear  equations  with  constant  coefficients  and  second 
member  zero  msiy  be  solved  as  follows : 
Substitute  in  the  given  equation, 

— izsm**,    ^  =  7^1**  S -^  =  m,    y  =  w/=l. 

dx*"  dx"-^  dx 

There  will  result  an  equation  of  the  nth  degree  in  m,  called 
the  auxiliary  equation.  Find  the  n  roots  of  this  equation ; 
these  roots  will  determine  a  series  of  terms  expressing  the 
complete  value  of  y  as  follows,  viz.  For  each  real  root  mj, 
there  will  be  a  term  Ce^r^ ;  for  each  pair  of  imaginary  roots 
a  ±  6V—  1,  a  term  e''''{Asmbx  -\-  Bcosbx)  ;  each  of  the  coeffi- 
cients A^  -B,  (7,  being  an  arbitrary  constant  if  the  corresponding 
root  occur  only  once,  but  a  polynomial  Ci  -|-  Cgo;  +  Cso;^  •  •  •  4-  c^ic''"^, 
if  the  root  occur  r  times. 


20  LINEAR   EQUATIONS. 

44.    Roots  of  auxiliary  equation^  real  and  unequal. 
Solve  the  following  equations  : 

dx^  dx 

3.       a^=^. 
doc^      dx 


4.     g(ty+y]=io'M. 

\  dx^       J  dx 

c         d^v  ,    .  dy 
dor        dx 


6. 


..(,+ii)=<„.+^,|. 


7.       ^  =  4^. 

g^        d^y^d^y  ^  ^dy 
dx^      dx^        dx 

dx^        dx 

11.        ^  «  2  (a^  +  62)  ^  +  (a^  -  by^  =  0. 
cZar       .  dor  dx 


45.    Boots  of  auxiliary  equation  unequal^  hut  not  all  real. 
Solve  the  following  equations  : 

1.     g+,  =  o. 

da?        dx 


LINEAR   EQUATIONS.  21 

3.  g_2a^  +  6>  =  0. 
dur  ax 

(1),  when  a>b;     (2),  when  a<b. 

4.  ^-iab^+{a'  +  b'ry  =  0. 

5.  g-21ogag+[l  +  (log«)T2/=0. 

6.  ^  +  2^  =  0. 

7.  ^  =  2/. 

8.  i^  +  ^  +  ^  =  3y. 
dai?       dm?      dx 


10. 


da;*         d»^ 


11.       ^  +  ia*y  =  0. 


46.   Auxiliary  equation  containing  equal  roots. 
Solve  the  following  equations  : 

1.       ^-.2a^  +  a22/  =  0. 
dx^  dx 


22  LINEAR  EQUATIONS. 

3.       ^  =  4^. 

diK^       dic^      die 

6.       ^  +  2n2|§  +  ^^2/  =  0.     • 
dx^  dor 

dx*         do;^         dic^      da; 

8.       ^_-4^+14^-20^+252/=0,  the  first 

da?'*         dar  dar  dx 

member  of  auxiliary  equation  being  a  perfect  square. 

9.    ^+^!:!^=o. 

dx""      dx*"-^ 

47.  Linear  Equations  with  constant  coefficients  and  second 
member  not  zero. 

There  are  two  methods  of  solution : 

First.  Method  of  Variable  Parameters.  —  Solve  the 
equation  by  Art.  43,  regarding  the  second  member  as  zero. 

Supposing  it  to  be  of  the  7ith  order,  this  value  of  y  will  con- 
tain n  arbitrary   constants.      Derive   from   it  the   successive 

dv    d^v       d"'~^v 

differential  coefficients,  -^,    — ^,  • ^  ;  then  differentiate  the 

dx    dx^        daf-^ 

values  of  v?  — »    — f  ? r-?  regarding  the  arbitrary  constants 

daj    dar       dx""'^ 

alone  as  variable,  and  place  these  n  results  equal  to  zero,  except 
the  last,  which  put  equal  to  the  second  member  of  the  given 
equation.  These  n  conditions  will  determine  expressions  for  the 
n  arbitrary  constants,  which  are  to  be  substituted  in  the  original 
expression  for  y. 


LINEAR  EQUATIONS.  23 

Second  Method. — By  successively  differentiating  the  given 
equation,  obtain,  either  directly  or  by  elimination,  a  new  differ- 
ential equation  of  a  higher  order  with  the  second  member  zero. 
Solve  this  by  Art.  43,  and  determine  the  values  of  the  super- 
fluous constants  so  as  to  satisfy  the  given  differential  equation. 
In  this  last  work  of  determining  the  superfluous  constants  all 
the  other  constants  may  be  regarded  as  zero. 


48.   Solve  the  following  equations  : 

1.  ^_7^  +  122/  =  a;. 
dx^        dx 

2.  ^-_2^  +  2^-2^  +  2/  =  a. 
dx'^         dx^         dx^         dx 

3.  ^^a'y^x+l. 
dx"         ^ 

4.  ^_2^  +  ^  =  e«. 
dx^         dx^      dx 

5.  g-a^^  =  a^. 

d^xf 

7.  — -^  —  2a-^  -f- a?y  =  e' ;     also  when  a  =  1. 
dx^  dx 

8.  — f  4-  n^y  =  cos  ax ;     also  when  a  —  n. 
dor 

^'       ^-5^  +  6^  =  6"*;     also  when  n  =  2,  or  n  =  3. 
doer         dx 

11.  ^^^^^20y  =  x'^'. 
dxr         dx 

12.  — f  -\-4y  =  x  sin^ic. 
dor 


24  LINEAR   EQUATIONS. 

13.  — f +  2— |-f-i/  =  a^cosai»;     alsowhena=l. 

14.  —J  — 2-^  +  42/  =  e*cosi». 

(XX/  (XX 

49.  Linear  equations  of  tlie  form 

where  -4i,  ^2)  '"-^n  ^-re  constants,  and  X  a  function  of  a?  alone. 

Put  o.  +bx  =  e\  and  change  the  independent  variable  from 
X  to  t.  The  new  differential  equation  between  y  and  t  will  be 
linear  with  constant  coefficients,  and  may  be  solved  by  Art.  47. 

50.  Solve  the  following  equations  : 

dxr        dx 
2.        {X  +  ay^  _  4 (oj  +  a)  ^  +  6y  =  x. 

3-        («  +  ^^y^  +  b{a  +  bx)^  +  hhj  =  0. 
dar  do? 

4.       a;2^  _  aj^  +  22/  =  x\ogx, 
da?        dx 

6.  i6(«,+  l)^^  +  96(a.  +  irg+104(..+  irg 

+  8(a;+l)^  +  2/  =  a^  +  4a!  +  3. 
da; 

7.  a,.^  +  6af'f|  +  9a;^g  +  3a;^  +  2/  =  (l+loga,r. 

do;*  dar  dar  aa? 

8.  x"^  -  (2m  -  1)0?^  +  W  +  n^)y  =  n^a^-loga?. 

dx^  dx 


CHAPTER  VIII. 

SOME    SPECIAL    FORMS    OF    DIFFERENTIAL    EQUATIONS    OF 
HIGHER   ORDERS. 

51.  Form,    -— ^  =  X,     where  X  is  a  function  of  x  alone. 

The  expression  for  y  is  found  by  integrating  X  successively 
71  times  with  regard  to  x.     Or  solve  by  Art.  47. 

52.  Solve  the  following  equations  : 

1.        x^=i. 


2. 


d'y_ 


dx'^      (ic-j-a)^ 

3.       ^=af. 
dx*^ 


4. 


d'y 

— ^  =  oleosa;. 

dx'^ 


5.  e"" — ^ -f- 4  cos  i»  =  0. 

dx' 

6.  ^  =  xe', 
dx"" 

7.  — ~  =  sin^ic. 
da^ 

(jpy 

53.   Form,    — ^  =  Y^    where  Y'  is  a  function  of  y  only. 
dor 

Multiplying  both  members  by  2-^,  and  integrating,  we  have 


(^=2  Crdy  +  c^,      Therefore   x=:  f ^ 


+  C2. 


26  EQUATIONS   NOT   CONTAINING  y. 

54.   Solve  the  following  equations : 

2.  ^  =  -aV 

3.  ^^^  =  a. 

4.  tl^^e^y. 


5. 


do^      -yjay 


55.  Equations  not  containing  y  directly. 

By  assuming  the  differential  coefficient  of  the  lowest  order  in 
the  given  equation  equal  to  z^  and  consequently  the  other  differ- 
ential coefficients  equal  to  the  successive  differential  coefficients 
of  z  with  respect  to  x^  we  shall  obtain  a  new  differential  equation 
between  z  and  a:  of  a  lower  order  than  the  given  equation. 

56.  Solve  the  following  equations  : 

1.  aj^  +  ^  =  0. 

dx^      dx 

2.  ^  =  a^J^h' 
da^ 


dy 
dx 


doer        \dxrj 

«■  "•©■--(IT 


5. 
6. 


EQUATIONS   NOT   CONTAINING  X. 


27 


8. 


10. 


doer  dx 

da^   dx"  [dx^J 

dx^  dx"     \  '    doj^yL       V^^y_ 
dx^  \dxj 


57.   Equations  not  containing  x  directly. 
By  assuming  -^=.z^  and  consequently 

^^z—,     ^^z'^^^  +  zf^^Y,    etc., 

changing  the  independent  variable  from  x  to  2/,  we  shall  obtain 
a  new  differential  equation  between  z  and  y  of  a  lower  order  than 
the  given  equation. 


58.   Solve  the  following  equations  : 


1. 
2. 
3. 
4. 
5. 
6. 


•  ^^^y)  -j3+(^  +  ^^Ey)(-~^ 


W'- 


dxr  \dxj 
2/(1 

dor  L\^^/  \darj  J       \dx_ 

\dx)  ''da?  dx  •'  \\dx)     dx"  j 


dx"' 


CHAPTER  IX. 

SIMULTANEOUS   DIFFERENTIAL  EQUATIONS. 

59.  Simultaneous  differential  equations  of  the  first  order. 
There  should  be  n  given  equations  between  n-{-\  variables. 

Selecting  one  of  these  for  the  independent  variable,  we  may,  by 
differentiating  the  given  equations  a  sufficient  number  of  times, 
eliminate  all  but  one  of  the  dependent  variables  and  their  differ- 
ential coefficients.  The  resulting  differential  equation  between 
two  variables  must  be  solved  by  the  methods  previously  given, 
and  from  its  primitive  and  the  given  equations  may  be  obtained 
the  values  of  the  other  dependent  variables.  The  complete 
solution  will  consist  of  n  equations  containing  n  arbitrary  con- 
stants. 

In  general,  if  we  differentiate  the  given  equations  71  —  1  times 
successively,  we  shall  have  in  all  n^  equations,  which  are  just 
sufficient  for  the  elimination  of  n  —  1  variables,  together  with 
their  n(n  —  l)  differential  coeflScients.  Shorter  processes  for 
the  elimination  will  frequently  suggest  themselves  in  special 
cases. 

60.  Solve  the  following  simultaneous  equations : 


1. 

!+*'+!-»• 

|  +  3,-«  =  0. 

2.       - 

ax-      dy      __      dz 

^y  +  4tz      2y-\-bz 

3.       - 

^^     _     -^y    ^dt. 

y  —  7x     2x-\-5y 


SIMULTANEOUS  EQUATIONS  OP  THE  FIKST  OBDEE.      29 


ax 


dx 


+  a2!=0. 


5. 


dt 


dy 
dt 


^3y  —  x=e^' 


dx 


____dy_ 


2y  —  ox  +  e*     x  —  6y-\-e^* 


=  dt. 


dt         dt 

3— +  7^  +  34  a;  +  382/ =  e*. 
dt         dt 


dt         dt 

S^^7^-\-Sx-{-24.y  =  e'K 
dt         dt  •" 


10. 


4— +  9^  +  2a;  +  3l2/  =  eS 
dt         dt 

3— 4-7^  +  a;  +  242/  =  3. 
^    dt         dt 

dt       t^       ^^ 
dt      t^  ^^ 


11. 


i; 


tdx=  {t  —  2x)dt, 
dy=z{tx-\'ty  +  2x  —  t)  dt. 


30      SIMULTANEOUS   EQUATIONS   OF  HIGHER  ORDERS. 


12. 


(  dx 
dt        ^ 

-^  =  lz'-  nx, 
dt 


=  mx  —  ly. 


13. 


f    dx 
lt—  =  mn{y-z), 
at 

7nt~^  =7il  (z  —  x). 
dt  ^         ^' 

nt  —  =  Im  (x  —  y) . 
dt  ^        ^^ 


61.  Simultaneous  differential  equations  of  an  order  higher 
than  the  first. 

By  differentiating  the  given  equations  a  sufficient  number  of 
times,  we  may  eliminate  all  but  one  of  the  dependent  variables 
and  their  differential  coefficients,  and  thus  obtain  a  differential 
equation  between  two  variables,  which  must  be  solved  by  the 
appropriate  methods.  Its  primitive,  together  with  the  given 
equation,  will  enable  us  to  determine  the  values  of  the  other 
dependent  variables.  The  general  solution  will  contain  a  num- 
ber of  arbitrary  constants  equal  to  the  sum  of  the  highest  orders 
of  differential  coefficients  in  the  several  given  equations. 


62.    Solve  the  following  simultaneous  equations  : 
1. 


dv 


—4^  —  n^x  =  0. 
dt^ 

d^x 


1 


df" 
d'y 

dt:- 


■Sx  —  4:y-\-S  =  0, 


^2  +x  +  y  +  5=:0. 


SIMULTANEOUS   EQUATIONS   OF  HIGHEB  OBDEES.        31 


—  -3a;- 42/ +  3  =  0, 

^+a;-8w  +  5  =  0. 
df  ^ 


4. 


(d'x   ,  .,2 


df 


+  n'y  =  % 


—  n^x  =  0. 


5. 


r2^-^-4y  =  2a;, 
daj^      do; 

2^  +  4— -32^  =  0. 
do;        dx 


dx' 


"  da^    '      do^ 
dx^        dx^ 


CHAPTER   X. 

GEOMETKICAL   EXAMPLES. 

63.   Expressions   involved   in  the  examples,  p  representing 

-^,  and  q  representing  — |. 
dx  dar 

y 

Subtangent=-.       Subnormal  =py. 

. ds  

Normal  =  2/ Vl  4-i>^.       "T~=VlH-p^. 

y 

Intercept  of  tangent  on  axis  oi  X=x 

Intercept  of  tangent  on  axis  of  Y=y—px. 

Radius  of  curvature  =  q:  -^ — JULJ—, 

Q 

1.  Find  the  curve  whose  subtangent  varies  as  (is  n  times)  the 

abscissa. 

2.  Find  the  curve  whose  subnormal   is   constant   and   equal 

to  2  a. 

3.  Find  the  curve  whose  normal  is  equal  to  the  square  of  the 

ordinate. 

4.  Find  the  curve  for  which   s  =  mx^, 

5.  Find  the  curve  for  w^hich   s^  =  y^  —  a^. 

The  orthogonal  trajectory  of  a  series  of  curves  is  a  curve 
that  intersects  them  all  at  right  angles. 

Describe  the  curves  represented  by  the  following  equations, 
and  find  their  orthogonal  trajectories  : 

6.  y  =  mx,      m  being  the  variable  parameter. 


GEOMETRICAL   EXAMPLES.  33 

7.  7f  =  2aa;  —  ic^,  a  being  the  variable  parameter. 

8.  y  =  4aa7,  a  being  the  variable  parameter. 
d,    xy  =  k^^                    k  being  the  variable  parameter. 

10. f-  ^  =z  1 ,  h  being  the  variable  parameter. 

11.    x^  -\- m^  if  =  m^  a^ ^    a  being  the  variable  parameter. 

12. h  — =  li      ^  being  the  variable  parameter. 

The  three  following  examples  require  the  singular  solution : 

13.  Find  the  curve  such  that  the  sum  of  the  intercepts  of  the 

tangent  on  the  axes  of  X  and  Fis  constant  and  equal  to  a, 

14.  Find  the  curve  such  that  the  part  of  the  tangent  betweei\ 

the  axes  of  X  and  Fis  constant  and  equal  to  a. 

16.  Find  the  curve  such  that  the  area  of  the  right  triangle» 
formed  by  the  tangent  with  the  axes  of  X  and  Y  is 
constant  and  equal  to  a?. 

The  following  examples  require -the  solution  of   differential 
equations  of  the  second  order  : 

16.  Find  the  curve  such  that  the  length  of  the  arc  measured 

from  some  fixed  point  of  it  is  equal  to  the  intercept  of 
the  tangent  on  the  axis  of  X, 

17.  Find  the  curve  whose  radius  of  curvature  varies  as  (is  n 

times)  the  cube  of  the  normal. 

18.  Find  the  curve  whose  radius  of  curvature  is  equal  to  the 

normal ;    first,  when  the  two  have  the  same  direction ; 
second,  when  they  have  opposite  directions. 

19.  Find  the  curve  whose  radius  of  curvature  is  equal  to  twice 

the  normal ;  first,  when  the  two  have  the  same  direction  ; 
second,  when  they  have  opposite  directions. 


AlsTSWERS. 


Art.  3.      (p  =  ^ 

dx 


) 


1.  y{l-{-x)-\-px{\-y)  =  0,     6.    y=px-j-2^-p\ 

2.  {x^-\-l)pxy  =  y^  +  l.  7.    xp^  =  a, 

3.  tan  a;  =  p  tan  2/.  8.  p2  + 2p2/cota;  =  2/^. 

4.  (l4-a;2)p+2/  =  tan-^a;.        9.    x'p^  =  l-\-p\ 

5.  (2/logic— 1)  ?/=pic. 

Art.   5. 

1.  ^4.a^2/  =  0.  4.    a.^^_a;^=3y. 
dx^  dor        dx 

2.  — ^  —  a'^y  =  0,  5.   — |  +  7i^2/  =  cos  aa;. 
daj2  dxr 

3.  ^-2a^  +  a^2/  =  0. 
dor  dx 

Art.   7. 
dx^         dx  '  *    dx^        dx^        dx^      dx 

dxr      dor      dx  dx* 

3.    ^_2^  +  ^  =  e^ 
dx^         dx^      dx 

Art.  9. 

1 .  log  (xy)  -^x  —  y^c.  t^ .    (1  +  x^)  (1  +  2/^)  =  cx^. 

2.  ?Jl^4.w^  =  c.  4-    xhj  =  ce^. 

xy    ^     ^x 


36 


ANSWERS. 


5.    log[(2/  +  Vl+2/')Vl+/] 


+  c. 


6.  cosy  =  cGosx. 

7.  tana;tan^  =  c. 

1.  y  =  ce~K 

2.  yz=ce-yl, 

3.  y  =  ce^, 

4.  x^  =  c^  +  2cy.Xj^\ 


8.  sin^  X+  sin^y  =  c. 

9.  xy—l  =  c{x-{-y  +  l). 

Art.   11. 

5—  sin? 
.      X  =  Ce  a;  . 

6.  {y  +  xy(y-{^2xy  =  c. 

7.  log  (aj2  +  /)  =  2  tan-i  -  +  c. 

2/ 


8.    xycos'-=c. 

X 


9.  (1) ,  log  (a;2  —  mxy  +  2/^)  + 


(2),  x  —  y  =  ce^~ 


2  m 


rtaii 


_i  2v  —  mx 

X  -y/4  _  -^2 


(3), 


(2y  —  mx  4-  a?V?7i^  —  4)" 


(22/  -  mx  -  a;Vm2-4)'"+^"^'"' 
10.   2/sina  — a;cosa  + Va^  +  2/^  =  c  (a^4-2/^)' 

Art.   13. 

1.  (^y  —  x-}-iy(y  +  x-iy  =  c. 

2.  x  +  2y-{-log{2x  +  y—l)=c. 

3.  x-\-5y  +  2=:c(x  —  y-\-2y. 

4.  4aj  — 82/  =  log(4aj4-82/  +  5) +c. 

6.   log[2(3a;-l)^+(3y-5)^]-V2taD-^'^^^^^~-^^=c. 

3y  —  5 


1.    y  =  cx''  + 


1—a      a 


ANSWERS. 
Art.   15. 
3.   2/  = 


37 


^  __4-ce^^. 


2.    y  =  ax-{-cx-\/l  —x^. 


4.   2/  =  sina;-l+ce-«^°". 


7.  2/vrT^  =  iog^^i-±^^^+<^- 


Art.   17 

3 


1.   y  =  {c-y/l-x'-a)-\ 
x-]-l       1 


2/=    ' 


ce2.^4.^(2a^  +  l)T^. 


2.   2/^  =  ce^ 


4.    ic  = 


5^    2/""'^^  =  ce^""^^"""'  +  2sina;  + 


2 


6.    y  =  {cx  +  \ogx  +  l)   \ 

8.    2/  = 


_(n±2)f  o     -      tana^  +  seco; 


a^r+'  +  c  \        sinx  +  c 

9.    (4&2  +  1)  2/2  =  2a(smaj  4-  2&C0SX)  +  ce^^^^ 


Art.   19. 

1.  x^-|-6a^2/'  +  2/'  =  c- 

2.  a^-Qx'y-exy'  +  f^c.      Q.   y'  =  c'-2cx. 

3.  x^  —  y^=cx. 

X 


5.    a^  +  2/^  +  2tan-i|  =  c. 


7.   a^  +  2/^  +  2sin-i-==c. 


88  ANSWERS. 

9.  e(p^^y^)^c, 

10.  cos  {mx  +  ny)  +  sin  {nx  =f  my)  =  c. 

11.  VTT^T?  +  tan-i2=c. 

12.  (1),  iog-:5!^^i±r^=2W%_^^^ 

(2),  tan-i^  +  ^!V6^  =  e. 

(3),  a^(^l_  lUe. 
Va      hy) 

(4),   ^^±lVf^,^.v^.^^ 

a      ^a;« 

Art.   22. 

1.  x'  +  y^  =  c{x  +  y).  4.    aj2  +  22/2=cV?T?. 

2.  aj2__^2^^^^^^  ^    y  =  cx. 

3.  2/  =  cflj. 


Art.    24. 


1 


1.    x=icye^.  4.   -4=  =  log  — 

2-  ^  =  ce^^  5.   xy=^c, 

3-  ^^-  — =  logc/.  6.    a;2/e^  =  log^. 


a; 


ca? 


7.    «2/  +  log  sin  (xy)  =  loff — 

y 

Art.  26. 

1.  e{^-^^)=.c.  3.   a^2_^2^^^^ 

2.  d^-.f^of.  4.   2/Vn^  +  a:(l-2/)=c. 


ANSWERS.  39 

5.    sinxcosy-{-ytsin2x  =  c.         6.    siii^a;sin2  (aj  —  ^/)  =  c. 
7.   y  =  e~J''"'ffQe-'''"''dx  +  c\ 

Art.   28. 

1.  {x^a)(y  —  b){z  —  c)  =  c'.      7.    z  =  c^. 

2.  a^+22/^— 6a?2/— 2i»2;+2;^=c.      8.    cc^/ =  <^ (^  +  ^) • 

3.  yZ'^zx  +  xy  =  c.  9.    a;  = f-c 

2/  +  ^ 

4.  xyz  =  c.  10.    2/(a?  +  2;)  =  c(2/  +  2;). 

5.  6=^(2/  + 2;)  =  c.  11.    e^(a;  +  2/  +  ^0  =  c. 

6.  ^  +  5  +  ?  =  c'.  12.   ^±_%^_±^  =  e. 
x     y     z  X  y 

Art.  30. 

1.  (i/-2aj  +  c)(?/-3a;  +  c)  =  0,    or    {bX'-2y -\- cy=^o?. 

2.  (?/  +  c)2  =  a2(loga;)2.  3.    {y-^cy  =  4.ax. 

4.  (2/  +  c)2  =  (  Va?  -  ar^  +  sin-^  VS)^ 

5.  {xy  +  c){o^y-Jrc)=0. 

6.  (x2-22/  +  c)[e^(aj4-2/-l)  +  c]=0. 

7.  (2/  +  c)(2/  +  ^  +  c)(ar2/  +  c2/+l)  =  0. 

8.  (x^-32/  +  c)(e2+c?/)(a;i/  +  c2/+l)  =  0. 

9.  2/2sin2a;4-2c2/  +  c2  =  0. 

Art.  32. 

1 .  Eliminate  p  by  means  of      x  =  — z=-  (c  -f-  a sin~^p) . 

Vi  — p- 

2.  Eliminate p  by  means  of      a;(p  —  1)^  =  logp^— 2p +  c. 

3.  Eliminate  p  by  means  of      a?  =  — ,  (  c  H f-  atan"^»  ). 


40  ANSWERS. 

4.  y  =  cx-^C'-c^. 

5.  a;  =  alog  {ay  ±  Va'  + 2/^—1)  +  log  (y  T  Va^  +  2/^  —  1)  +  c. 

6.  a;±  Vc?+46^  =  alog(a±  Va2  +  46?/) +c. 

w       /«     L       /-2-^ — \^i7      d=4  Vi«^  +  ^  — i:c(  Vl7  — 1) 

7.  c(22/±a;Va;2  +  2/)      = ^    , -. 


y  +  V?/^  —  of 


=  (2/2  +  a^logca^)2 


8.  2/V/-a^-a^log-  ^ 

9.  4(a;  +  c)3+(x  +  c)2-182/(a;  +  c)  -  272/'- 4^/  =  0. 

Art.   34. 

1.    y^  =  2cx  +  c^. 


2.  0^  +  1  =  ±  V22/  +  c  +  log(±  V22/  +  C-1). 

1  y 

3.  (e«  — ac)2r=2caje«. 

4.  e-^  +  2ca;e^  +  c2=0. 


a''  —  1 
when    a=l,      4^/  =  a;'  — log  (ca^). 
6.    62(62/  +  c)2+(6a&aj  +  a3)(62/  +  c)-3a2a;2-16&aj3  =  0. 


7.    c(naj2+22/2±ajVnV+4m2/2)»^=  [(2m~n)a;± VnV+4m2/']^ 

Art.   36. 

1.    (a^  -  2/2  +  c)  (0^2  -  2/' +  c^')  =  ^• 

3.  3a;*  +  6ca^2/  +  ^'  =  ^- 

4.  (2/-cc)^^^2^c(V^  +  V^)l 

6.    x^+2cx*-iy-o27i  =  0,     where  A  =  ^^- 


ANSWERS.  41 


Art.    38. 


2.  y  z=cx-{-c  —  c^.  6.    2/^  =  cx^  -\-  (?, 

3.  (2/  — ca;)2=  l+c^.  7.    e^=ce^  +  c"\ 

4.  2/'=cx  +  — •  8.    y^-cx^-       ^^'' 


c-hl 


Art.   42. 
Complete  Primitives  and  Singular  Solutions : 


1. 

y  =  cx  +  ^, 

C 

2/2  =  4  mo;. 

2. 

(y  —  cxy==  1— c^, 

f^^  =  \. 

3. 

y  =  c(x-cy, 

y='-\ 
^      27 

4. 

y=ic{x-\)-(?. 

42/=:(aj~l)2. 

5. 

2/2_2caj  +  c2  =  0, 

f=^x\ 

6. 

(2/~ca^)2+4c  =  0, 

xy=l. 

7. 

{y  —  cic)  (mc  —  n)  =  mnc^ 

\mj        \nj 

Art. 

44. 

ax  hx 

2.  2/  =  Cie2*  +  C2e**.  6.   2/  =  Ci^^  +  Cae". 

X 

3.  2/  =  Cie«  +  C2.  7.   2/  =  Cie2^  +  C2e-2*+C3. 

4.  y  =  Cie^'' -\- C2eK  8.   2/  =  ^i^^"" +  026-2^  +  03. 

9.   y  =  Cie^''-\-C2e~^''-^Cse', 

10.  2/  =  Cie^*  +  026"^="  +  636=^^3  +  646-^^^ 

11.  2/  =  Oie^"-^^^  +  Cge^*-"^^  +  C3e^«+^>^  +  C4e-(«+^>*  +  C5. 


42  ANSWERS. 

Art.   45. 

1.  2/  =  CiSina;  +  C2COsa?. 

2.  2/  =  e^*(ciSin2a?4-C2COs2aj). 

3.  When  a  >  6,     3/  =  e«*(cie^v^^  +  C2e-^V^^-^')  5 


when  a<hj     y=e'^(cismx V6^  —  o?-\-C2 cos x V^^  —  o?) . 
4.2/  =  e^"^*  [ci  sin  (a^  -  Z?^) a;  +  Cg cos  (a^  -  W) x'] . 

5.  2/  =  a*(cisina;  +  C2C0sa;). 

6.  2/  =  Ci  sin  a;  V2  +  C2C0sa?  V2  +  C3. 

-*  /         a;  "v/S  a;  "v/3\ 

7.  2/  =  Cie='4-e  ^^sin-^— +  C3eos-^j. 

8.  2/6""  =  Ci e^*  +  C2 sin  x  V2  +  C3 cos  x  V2. 

9.  2/  =  Cie*  +  C2  6~*  +  C3sina;-f  C4eosaj. 

10.  y  =  Cie*^2  _)_  c2e-*^2  +  Cssin  2  a:  +  C4COS  2  aj. 

11 .  2/  =  e*"  (ci  sin  ax  +  Cg  cos  ax)  +  e"***  (cg  sin  ax  +  C4  cos  ax) . 

12.  2/ =  Cie*  +  C2e-*  + (0362  +  046  2)  sin ^^ 

a;V3 


+  (0562 +  C6e  2)  cos 


13.  2/  =  Ci  sinoj  +  C2  cos  x-\-e  ^  (  C3  sin  -  +  C4  cos  -  j 

I  ^ — 2~i      .   aJ  ,  a;\ 

+  e    2 /c5Sin-  +  C6Cos-j. 

14.  y  =  Cie="+  026"*+  Cgsina;  +  C4COsa;+e^Y^5sin—  +C6Cos— ) 

V         V2  V2/ 


+  e  ^^2/'c7sin-^  +  C8Cos-^Y 
\         V2  V2y 


Art.   46. 

1.   2/=  (ci  +  C2aj)e«'.  2.   2/  =  Ci  +  C2a;. 

3.   2/  =  Cie*'  +  C2  +  C3a;.  4.   2/  =  ^1^-*+ (02  +  033;) e^' 


ANSWERS.  48 

6.    y  =  (Cj  4-  ^2^)  cosnx  +  (cg  +  c^x)  sinnx, 

8.  ?/  =  e=^  [ (cj  +  Cgic)  sin  2 aj  +  (cg  +  040?)  cos  2  a;] . 

9.  y  =  Ci-\-C2X-{'CsX^  '"  +Cn_2X''~^  +  c^^iS\nx  +  c^cosx, 


1.   2/  =  Cie3=«  +  C2e^^-+ 


Art.   48. 

12a;+7 


144 

2.  2/  =  CiSina;  +  C2COsaj  + (C3  + C4a;)e'-fa. 

3.  2/  =  Cie«'  +  C2e-«*  — ^-i-i. 


0.    y  =  CiS'^' -\- CiB  ""' -\-c^^\nax-\-c^Q,osax • 

a* 

0.    2/ =  Ci  sin  Tix  +  C2  cos  no?  +  ' 


7.   2/  =  (Ci  +  C2a;)e«*  + 


when    a  =  l,     2/=[^i  +  ^2^H je^ 

8_           .          ,                  ,    cos  aa; 
.    y  =  Ci  sin  na;  +  C2  cos  nx  -|-  — ; 


X  sin  7tx 
when    a  =  71,     2/  =  ClSln^i»^-C2Cos?^a;  +  - 


9.   2/  =  Cie2-  +  c2e3«-|- 


2ri 


when   n  =  2,     2/  =  (^1  — ^)6^''  +  C2  6^*; 
when  71  =  3 ,     2/  =  ^i  ^^'^  +  (^2  +  ^0  6^''- 
10.   y^c,e^  +  c,e^'  +  -J;^ (2n-3)e'" 


44  ANSWERS.  • 

11.  y  =  c,e''  +  c,e"  +  2^^  +  6a;  +  7 ^.^ 

12.  y=U-^^^m'ix  +  (c,-^co^2x  +  '^. 

13.  3/  =  (Ci  +  C2a;)  siiia;  +  (c3  +  C4a;)  cosa; 

1  smaj 


I  cos  a:;. 
^x\ 


Art.  50. 

a? 


1.   2/  =  Cia^+  ^ 

3.  2/  =  Ci sin  log  (a  +  hx)  +  Cg cos  log  (a  +  5a3)  * 

4.  2/  =  ^(CiSiiiloga;  +  C2COsloga7  +  logaj). 

5.  2/  =  (2a;-l)[ci  +  C2(2a!-l)2  +C3(2a;-1)    ^]. 

6.  y  =  [.,  +  c.  log  (X  +  1)]  V^Tl  +  '^  +  '^  !°^.^^,  ^  ^^ 

\lx  +  1 


225 

7 .  2/  =  (^1  +  ^2  log  a;)  sin  log  oj  +  (cg  +  C4  log  a;)  cos  log  x  +  (log  a;)  * 

+  21oga;-3. 

8.  y  =  af"(ci sin logx**  +  Cgcos logo;"  +  logo;) . 

Art.  52. 

1.    2/  =  Ci  +  CaOJ  +  Cga^^  +  a^logaj. 


2.   2/  =  Ci  +  C2a?  +  C3a:^  +  C4X^  — (aj-f  a)^log Vic  +  a. 


ANSWERS.  45 


3.   y  =  Ci  +  C2X -{- CqX^  " •  -{-  c,, x^~^  + 


m  -\-7i 

4.  2/  =  Ci-{-C2X'i-CsX^  -{-c^a^  -{-xcosx  —  4  sin  a;. 

5.  y  =  Ci  +  C2X-{-CsxF -{-c^x^ -{-e~'cosx. 

6-   2/  =  ^1  +  ^2^  +  CgOJ^*.*  +  c„aj"~^  +  (a;  —  n)e'^- 

Art.   54. 


1.    ax  =  \og{y  +  -\/y^-{-Ci)-^C2,    or   ?/==  c'ie'^*  + c'ae""*. 


-1^ 

Ci 

3.    (c,x  +  C2y  +  a=c,y\ 


2.    ax  =  sin  ^^  -f-  Cg,    or  y  =  Ci  sin  (ax  +  C2) . 

Ci 


4.  xV2¥=c,log^^g^^^^+"^-^+c. 

5.  Sx=2a'(y^  -^2c,)(y^  +  c,y  +C2. 


Art.   56. 

1.  y  =  Ci\ogx  +  C2. 

2.  6^2/  =  log  sec  [a&(a;  +  Cj)]  +  Cg. 

3.  y  =  '-l^  +  xf(cO+C2. 

a 

5.  (x  +  c0^4-(2/  +  C2)'  =  ^'. 

6.  2/  =  CiX+ (ci2+l)log(x-Ci) +C2. 

7.  2/  =  Cisin~^a:  'r  (sin~^a;)^  +  C2. 

4  - 

8.  y  =  -—(x  +  c,^a^)^-{-C2X-\-Cs. 


46  ANSWERS. 

9.    l2y  =  w^  +  Citv  —  6wlogW'j'C2,     where   w  =  x  +  Cs, 

10.    2  2/ V^=  w  -y/w'  +  Ci^  +  c/log  (w;  -I-  Vti?Tc?)  +  Cj, 
where   w  =  x  -\-  c^. 

Art.   58. 

1.  y'  =  x'  +  c^x  +  C2.  4.    Cii»=:plog[ci2/+/(ci)]+C2. 

2.  log?/-l  = .  5.    log2/  =  Cle^  +  C2  6-^ 

^1  "^  "T"  ^2 

3.  c^y  =  C2e''i^  —  VT+oFc^K       6.    2/^+^  =  Cie''^*  +  Cg. 

Art.   60. 
(2x  =  (2c2  —  Ci  —  C2t)e   ^, 


2  a;  =  e~^'[  (ci  +  Cs)  sin  ^  +  (cg  —  Ci)  cos  i] , 


g      |2aj  =  e-^'L(Ci  +  C2 
1?/  =  e-6*(ClSm^^- 


■  C2  cos  ^) 
'2/  =  Cie"*  +  C2e-' 


az  =  —  nci  e""""  +  ^Cg  e 


n^-l 


6. 


^  '  ^  36      25 

J/ =  -  (c,  +  C2  4- C20e-^' +  ^V -• 

27      40' 

y=:c,e''^  +  C2e-'^  +  l^  +  ^. 
54:        40 


ANSWERS. 


47 


,  ,        _6,      29e'  ,  19^      56 
r     ,      .\  -At      49  e2*      31  e' 


36 


2/  =  -(Ci  +  C2  +  C20e~*'  + 


25  ' 
19  e^^      lie' 


36 


25 


9. 


10. 


(.       .     ,  ,\      _4/     ,     Ox  S  \jO 

Cismt  +  c^cost)  e  ^^ +  __-_, 

Zb         1  / 


2e' 


J  =  [(^2  -  Ci)  sin^  -  (C2  +  Ci)  cos^]  e"*'  "J^'^Jj' 

10      15 
^  '  '  20       15 


11. 


'x  =  Cit-^+-, 


y^c^e'-c^t  2__. 


12.  ^  2/  =  ^1  sin  ^t  4-  &2  cos  A:^  4-  ?>3, 
2;  =  Ci  sin  A:^  +  C2  cos  A;^  +  C3, 
where   A:^  =  Z^  +  m^  +  7^^ 


The  arbitrary  constants  are  connected  by  the  following  equa- 
tions : 

mci  —  nhi  _  nai  —  hi  __  Ibi  —  maj  _  , 

0/2  O2  C2 

«      m     n 


48  ANSWERS. 

{X  =  ajsin  {h\ogt)  +  agcos  {klogt)  +  ag, 
2/  =  &i  sin  (/clogO  +  &2  cos  (A:logO  +  ^s? 
z=.Ci  sin  (/clog^)  +  C2  cos  (A:log^)  +  C3, 

where  A;^  =  Z^  +  m^  +  n^. 

The  arbitrary  constants  are  connected  by  the  following  equa- 
tions : 

mn  (ci  —  &i)  _  7i?(ai— Ci)  __  Im  (61  ~  %)  _.  j^ 

?ai  +  m^6i  +  n^Ci  =  0,       a^=^  h^  =03. 


1. 


2. 


3. 


Art.  62. 

'  flj  =  Ci  sin  nt  -f  Cg  cos  n^, 


{:: 
{ 


aj  =  (ci  +  C2O  e'  +  (C3  +  C4O  e-'  -  23, 

-  2 2/  =  (ci  -  C2  +  C2O e*  +  (C3  +  C4  +  c^O^"'  —  36. 

€/      .     n?5    ,  ^i\  ,    -^/     .     nt  nt\ 

To/      .      nt  nt  \  ,    -%/  .     ?i^    ,  ^^^^  \ 


r  ii/  =  (ci  +  c^x)  (f  +  Scse 
5.    f 


T_?, 


6.    i  _3^     1 

U  =  2(3c2  — Ci— C2i:c)e='  — Cge   ^""3* 

6.       y  =  u  +  v^     z  =  —  u  +  v^ 

where     ?^  =  c^  e^-  +  (cg  e-*  +  C3  e""')  e«, 


ANSWERS.  49 

where       ii  =  (ci  +  C2X  +  c^e"^^  +  c^e-'''^^)e% 

Art.  63. 

1 .  a?  =  C2/**. 

2.  2/^  =  4  aa;  +  c,       a  parabola. 


3.  ±(a;  +  c)=log(2/  +  V/-l), 

or  2/  =  i(e''"^''  +  e'^  ''j ,        a  catenary. 

4.  4  m^/  -f-  c  =  2  mx  ■\/4m^a^—  1  +  log  (2  mo;  —  V4  m^a^  —  1) . 


5.    ±{x-\-  c)  =  alog  {y  +  V^/^  —  a^) , 

or  ±{x  +  c)=  aXo^y  +  ^^'  ~  ^^ 


a 


from  which     ^/zn-le^+e     "j,       a  catenary. 

6.  a;^  4-  2/^  =  c^,        a  circle. 

7.  ic^  +  2/^  —  2  C2/  =  0,        a  circle. 

8.  2  0^  +  2/^  =  2  c2,        an  ellipse. 

9.  :k?  —  y'^z=c^^        an  equilateral  hyperbola. 

10.  2/^  +  ic^  =  a21oga;^  +  c. 

11 .  y  =  cx^^, 

X^  ^2 

12.  — ^  — -^  =1,        an  ellipse  or  hyperbola. 

/r  —  c^& 

13.  a;^  +  2/^  =  ot^,        a  parabola. 

14.  x^  -\-y^  =  a^^        a  hypocycloid. 

16.    2xy=^a^^       an  equilateral  hyperbola. 
16.   c^2/^  — log 2/^  =  4 c(a;  +  c'). 


50  ANSWERS. 

17.  Gif" {x-\-  c'y=  1,    a  hyperbola,  when  n  >  0  ;  an  ellipse 

n 

or  hyperbola,  when  n  <  0. 

18.  Fu-st,  (x  +  c')2  +  2/2  =  C-,    a  circle. 
Second,  ±(x-i-c')  =  c  log  {y  +  V^/^  —  c^) , 


a  catenary. 


19.    First,  x-\-c'  =  c  vers  ^ '\/2cy  —  2/^,        a  cycloid 

Second,         (a;  +  c') ^  =  2  ci/  —  c^,        a  parabola. 


or                ±(a;  +  c')-clog^+^^'" 

c 

from  which     y  =  -le  '^    -{-e     "    L 
2\                      / 

?/ 

a  ( 

MATHEMATICS.  81 

Wentworth's  College  Algebra. 

By  G.  A.  Wentworth,  recently  Professor  of  Mathematics,  Phillips 
Exeter  Academy.  Half  morocco.  500  pages.  Mailing  price,  $1.65 ;  for 
introduction,  $1.50.  Answers  in  pamphlet  form ,  /ree,  on  teachers^  orders. 

rPHIS  is  a  text-book  for  colleges  and  scientific  schools.     The 

first  part  is  simply  a  concise  review  of  the  principles  of 

Algebra  preceding  quadratics,  with  enough  examples  to  illustrate 

and  enforce  the  principles.     The  work  covers  a  full  year,  but  by 

omitting  the  starred  sections  and  problems,  the  instructor  can 

arrange  a  half-year  course. 


William  Beebe,  Assistant  Profes- 
sor of  Mathematics  and  Astronomy ^ 
Yale  University:  I  find  it  charac- 
terized by  the  clearness  and  method 


of  all  Professor  Wentworth*s 
books,  and  am  particularly  struck 
with  the  amount  of  matter  in  the 
Algebra. 


Wentworth's  Elements  of  Algebra. 

By  G.  A.  Wentworth.  Half  morocco.  x+  325  pages.  Mailing  price, 
$1.25 ;  for  introduction,  $1.12.  Answers  hound  separately  in  pamphlet 
form. 

nPHIS  book  is  designed  for  high  schools  and  academies,  and 

contains  an  ample  amount  for  admission  to  any  college. 

Wentworth's  Complete  Algebra. 

By  G.  A.  Wentworth.  Half  morocco.  525  pages.  Mailing  price, 
$1.55 ;  for  introduction,  $1.40.  Answers  hound  separately  in  pamphlet 
form. 

rPHIS  work  consists  of  the  author's  Elements  of  Algebra,  with 

about  one  hundred  and  eighty-five  pages  additional. 

Wentworth's  Shorter  Course  in  Algebra. 

By  G.  A.  Wentworth.  Half  morocco.  258  pages.  Mailing  price, 
$1.10 ;  for  introduction,  $1.00.  Answers  in  pamphlet  form^  free,  on 
teachers'  orders. 

nPHIS  book  is  based  upon  the  author's  Elements  of  Algebra,  but 

with  fewer  examples,  so  as  to  make  a  one-year  course. 

Algebraic  Analysis. 

By  G.  A.  Wentworth  ;  J.  A.  McLellan,  Inspector  of  Normal  Schools, 
Ontario,  Canada;  and  J.  C.  Glashan,  Inspector  of  Public  Schools, 
Ottawa,  Canada.  Part  I.  concluding  with  Determinants.  Half  leather. 
x  +  418  pages.  Mailing  price,  $1.60;  to  teachers  and  for  introduction, 
$1.50. 


82  MATHEMATICS. 

Wen f worth's  New  Plane  Geometry. 

By  G.  A.  Wentworth,  recently  Professor  of  Mathematics,  Phillii)8 
Exeter  Academy.  12mo.  x  +  242  pages.  Mailing  price,  85  cents ;  for 
introduction,  75  cents. 

Wentworth' s  New  Plane  and  Solid  Geometry. 

By  G.  A.  Wentworth.  12mo.  Half  morocco,  xi  +  386  pages.  Mailing 
price,  $1.40;  for  introduction,  $1.25.  The  book  now  includes  a  treatise 
on  Conic  Sections  (Book  IX.). 

A  LL  the  distinguishing  characteristics  of  the  first  edition  have 

been  retained.    The  subject  is  treated  as  a  branch  of  practical 

logic,  the  object  of  which  is  to  detect  and  state  with  precision  the 

successive  steps  from  premise  to  conclusion. 

In  each  proposition  a  concise  statement  of  what  is  given  is 
printed  in  one  kind  of  type,  of  what  is  required  in  another,  and 
the  demonstration  in  still  another.  The  reason  for  each  step  is 
indicated  in  small  type  between  that  step  and  the  one  following; 
and  the  author  thus  avoids  the  necessity  of  interrupting  the  process 
of  demonstration  to  cite  a  previous  proposition.  The  number  of 
the  section  on  which  the  reason  depends  is,  however,  placed  at  the 
side  of  the  page;  and  the  pupil  should  be  prepared,  when  called 
upon,  to  give  the  proof  of  each  reason.  Each  distinct  assertion  in 
the  demonstrations  and  each  particular  direction  in  the  construc- 
tion of  the  figures  begins  a  new  line,  and  in  no  case  is  it  necessary 
to  turn  the  page  in  reading  a  demonstration. 

In  the  new  edition  will  be  found  a  few  changes  in  the  order  of 
the  subject-matter.  Some  of  the  demonstrations  have  been  given 
in  a  more  concise  and  simple  form.  The  diagrams,  with  which 
especial  care  was  taken  originally,  have  been  re-engraved  and  mate- 
rially improved.  The  shading,  which  has  been  added  to  many  of 
the  figures,  has  proved  a  great  help  to  the  constructive  imagination 
of  pupils.  The  theory  of  limits  —  the  value  of  which  the  author 
emphasizes  —  has  been  presented  in  the  simplest  possible  way,  and 
its  application  made  easy  of  comprehension. 

But  the  great  feature  of  this  edition  is  the  introduction  of  nearly 
seven  hundred  original  exercises,  consisting  of  theorems,  problems 
of  construction,  and  problems  of  computation,  carefully  graded  and 
adapted  to  beginners  in  Geometry. 


MATHBMATICa  83 

Wentworth's  Trigonometries. 

By  G.  A.  Wentworth. 
Plane  and  Solid  Geometry,  and  Plane  Trigonometry. 

12mo.     Half  morocco.     490  pages.     Mailing  price,  $1.55;  for  intro- 
duction, $1.40. 

Mew  Plane  Trigonometry. 

12mo.    Paper.    134  pages.    Mailing  price,  45  cents ;  for  introduction, 
40  cents.    The  old  edition  is  still  issued. 

New  Plane  Trigonometry,  with  Tables. 

8vo.    Cloth.     249  pages.    Mailing  price,  $1.00;  for  introduction,  90 
cents.    The  old  edition  is  still  issued. 

I\lew  Plane  and  Spherical  Trigonometry. 

12mo.    Half  morocco.    214  pages.    Mailing  price,  95  cents ;  for  intro- 
duction, 85  cents.    The  old  edition  is  still  issued. 

li/ew  Plane  and  Spherical  Trigonometry,  with  Tables. 

8vo.   Half  morocco.    315  pages.   Mailing  price,  $1.30;  for  introduction, 
$1.20.    The  old  edition  is  still  issued. 

New  Plane  Trigonometry,  and  Surveying,  with  Tables. 

8vo.    Half  morocco.    305  pages.    Mailing  price,  $1.35;  for  introduc- 
tion, $1.20. 

New  Plane  and  Spherical  Trigonometry  and  Surveying,  with  Tables. 
8vo.    Half  morocco.    368  pages.    Mailing  price,  $1.50;  for  introduc- 
tion, $1.35. 

New  Plane  and  Spherical  Trigonometry,  Surveying,  and  Navigation. 

12mo.    Half  morocco.    412  pages.    Mailing  price,  $1.30;    for  intro- 
duction, $1.20. 

rpHE  aim  has  been  to  furnish  just  so  much  of  Trigonometry  as 

is  actually  taught  in  our  best  schools  and  colleges.     The 

principles  have  been  unfolded  with  the  utmost  brevity  consistent 

with  simplicity  and  clearness,  and  interesting  problems  have  been 

selected  with  a  view  to  awaken  a  real  love  for  the  study.     Much 

time  and  labor  have  been  spent  in  devising  the  simplest  proofs  for 

the  propositions,  and  in  exhibiting  the  best  methods  of  arranging 

the  logarithmic  work.     Answers  are  included. 

The  New  Plane  Trigonometry  gives  sufficient  practice  in  the 
radian  as  the  unit  of  angular  measure,  in  solving  simple  trigono- 
metric equations,  in  solving  right  triangles  without  the  use  of 
logarithms,  and  in  solving  problems  in  goniometry. 

It  also  contains  the  latest  entrance  examination  papers  of  some 
of  the  leading  colleges  and  scientific  schools ;  and  a  chapter  on 
the  development  of  functions  of  angles  in  infinite  series. 


84  MATHEMATICS. 

The  New  Spherical  Trigoiiometry,  Surveying,  and  Navigation 
has  been  entirely  re-written,  and  such  changes  made  as  the  most 
.recent  data  and  methods  seemed  to  require. 

Cooper  D.  Schmitt,  Professor  of 
MathematicSy  University  of  Tennes- 
see, EnozviUe,  Tenn. :  For  a  short 
course  and  quick  learning  of  the 
practical  application  of  the  subject,  I 
heartily  commend  Wentworth's  New 
Plane  and  Spherical  Trigonometry. 


W.  P.  Durfee,  Professor  of  Mathe- 
maticSf  Hohart  College,  Geneva, 
N.Y.:  I  have  examined  Wentworth's 
New  Trigonometry  and  think  it  an 
improvement  of  an  already  excellent 
book. 


Wentworth  &  Hill's  New  Fiue-Place  Logarithmic 

and  Trigonometric  Tables, 
By  G.  A.  Wentworth,  and  G.  A.  Hill. 

Seven  Tables  (for  Trigonometry  and  Surveying) :  Cloth.  8vo.  79  pages. 
Mailing  price,  55  cents ;  introduction,  50  cents. 

Complete  (for  Trigonometry,  Surveying,  and  Navigation) :  Half  mo- 
rocco.   8vo.    XX +  154  pages.    Mailing  price,  $1.10;  introduction,  Sl.OO. 

nPHESE  Tables  have  been  prepared  mainly  from  Gauss's  Tables, 
and  are  designed  for  the  use  of  schools  and  colleges.  They 
are  preceded  by  an  Introduction,  in  which  the  nature  and  use  of 
logarithms  are  explained,  and  all  necessary  instruction  given  for 
using  the  tables.  They  are  printed  in  large  type  with  very  open 
spacing.  Compactness,  simple  arrangement,  and  figures  large 
enough  not  to  strain  the  eyes,  are  secured  by  excluding  propor- 
tional parts  from  the  tables. 

Wentworth's  Analytic  Geometry. 

By  G.  A.  Wentworth.    Half  morocco.    301  pages.    Mailing  price, 

$1.35 ;  for  introduction,  $1.25. 
rpHE  aim  of  this  work  is  to  present  the  elementary  parts  of  the 
subject  in  the  best  form  for  class-room  use.  The  exercises 
are  well  graded,  and  designed  to  secure  the  best  mental  training. 
By  adding  a  supplement  to  each  chapter,  the  author  has  made 
provision  for  a  shorter  or  more  extended  course. 

Wentworth's  Logarithms  and  Metric  Measures. 

By  G.  A.  Wentworth.    12mo.     Paper.    61  pages.     Mailing  price, 
26  cents ;  for  introduction,  20  cents. 


MATHEMATICS.  85 

Wentworth  &  HiU's  Exercises  in  Arithmetic. 

I.  Exercise  Manual.  12mo.  Boards.  282  pages.  Mailing  price,  65 
cents;  for  introduction,  50  cents.  II.  Examination  Manual.  12mo. 
Boards.  148  pages.  Mailing  price,  40  cents ;  introduction  price,  35  cents. 
Both  in  one  volume^  80  cents.    Answers  to  both  parts  together^  10  cents. 

^HE  Exercise  Manual  contains  3869  examples  and  problems 

for  daily  practice.     The  Examination  Manual  contains  300 

examination-papers,  progressive  in  character. 

Wentworth  &  Hill's  Exercises  in  Algebra. 

I.  Exercise  Manual,  12mo.  Boards.  232  pages.  Mailing  price,  40 
cents;  for  introduction,  35  cents.  II.  Examination  Manual.  12mo. 
Boards.  159  pages.  Mailing  price,  40  cents ;  for  introduction,  35  cents. 
Both  in  one  volume^  70  cents.    Answers  to  both  together^  25  cents. 

rrHE  first  part  contains  about  4500  carefully  arranged  problems 

in  Algebra.     The  second  part  contains  nearly  300  progressive 

examination-papers. 

Wentworth  &  Hill's  Exercises  in  Geometry. 

12mo.  Cloth.  255  pages.  Mailing  price,  80  cents;  for  introduction, 
70  cents.    Answers  are  included  in  the  volume, 

rPHE  exercises  consist  of  a  great  number  of  easy,  carefully  graded 
problems  for  beginners,  and  enough  harder  ones  for  more  ad- 
vanced pupils. 

Wentworth  "&  Hill's   Examination    Manual  in 

Geometry. 

12mo.    Cl( 
60  cents. 

Wentworth' s  Geometrical  Exercises. 

By  G.  A.  Wentworth.  12mo.  Paper.  64  pages.  Mailing  price,  12 
cents:  for  introduction,  10  cents. 

A   SERIES  of  exercises  exactly  parallel  to  those  of  Wentworth's 

New  Plane  and  Solid  Geometry. 

Wentworth's  Syllabus  of  Geometry. 

By  G.  A.  Wentworth.  12mo.  Paper.  60  pages.  Mailing  price,  27 
cents ;  for  introduction,  25  cents. 

rPHIS  Syllabus  contains  the  captions  of  the  propositions  in  Went- 
worth's Plane  and  Solid  Geometry,  numbered  as  in  the  book. 


12mo.    Cloth.    138  pages.    Mailing  price,  55  cents  ;  for  introduction, 
60  cents. 


86  MATHEMATICS. 

Hill's  Geometry  for  Beginners. 

By  G.  A.  Hill.    12mo.    Cloth.    320  pages.    Mailing  price,  $1.10 ;  for 
introduction,  $1.00.  Answers^  in  pamphlet  form  ^  can  he  had  by  teachers. 
rPHIS  book  presents  the  subject  in  the  natural  method  as  distin- 
guished from  the  formal  method  of  Euclid,  Legendre,  and  the 
common  text-books.     The  central  purpose  is  intellectual  training, 
or,  teaching  by  practice  how  to  think  correctly  and  continuously. 

Hill's  Lessons  in  Geometry. 

For  the  Use  of  Beginners.  By  G.  A.  Hill.  12mo.  Cloth.  190  pages. 
Mailing  price,  75  cents ;  for  introduction,  70  cents.  AnmverSf  in  pam- 
phletform^  can  he  had  hy  teachers. 

rPHIS  is  a  course  similar  to  that  given  in  the  Geometry  for 

Beginners,  but  it  is  shorter  and  easier,  and  does  not  require 

a  knowledge  of  the  metric  system. 

Hill's  Drawing  Case. 

Prepared  expressly  to  accompany  Hill's  Lessons  in  Geometry,  and  con- 
taining, in  a  neat  wooden  box,  a  seven-inch  rule  with  a  scale  of  milli- 
meters ;  pencil  compasses,  with  pencil  and  rubber ;  a  triangle ;  and  a 
protractor.    Retail  price,  40  cents ;  for  introduction,  30  cents. 

A  specimen  copy  of  the  Lessons  in  Geometry  with  the  Drawing  Case 
will  be  sent,  postpaid,  to  any  teacher  on  receipt  of  $1.00. 

Gay's  Business  Book-Keeping. 

By  George  E.  Gay,  Superintendent  of  Schools,  Maiden,  Mass.  Quarto. 
Cloth.  Printed  in  red  and  black,  with  illustrations  and  finely  engraved 
script. 

Single  Entry  (Grammar  School  edition).  Quarto.  93  pages.  Mailing 
price,  75  cents ;  for  introduction,  66  cents. 

Double  Entry.  Quarto.  142  pages.  Mailing  price,  $1.25 ;  for  intro- 
duction, $1.12. 

Complete  (High  School  edition).  Quarto.  226  pages.  Mailing  price, 
$1.55;  for  introduction,  $1.40. 

Blanks,  money,  and  merchandise  are  provided.    Send  for  full  descrip- 
tive circular. 
nnniS  work   is   a  concise,   teachable   manual   of   the   modem 
methods  of  recording  business  transactions. 

Algebra  Reviews. 

By  Edward  R.  Robbins,  Master  in  Mathematics  and  Physics,  Law- 
renceville  School,  Lawrenceville,  N.J.  12mo.  Paper.  44  pages.  Mail- 
ing price,  27  cents ;  for  introduction,  25  cents. 

T^HIS  little  book  is  intended  to  be  used  only  during  review  and 

in  place  of  the  regular  text-book  in  elementary  algebra.     A 

list  of  eleven  recent  college  examinations  has  been  added. 


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